New Sasaki-Einstein 5-manifolds

We prove that closed simply connected $5$-manifolds $2(S^2\times S^3)\mathrm{\#}n{M}_2$ allow Sasaki-Einstein structures, where $M_2$ is the closed simply connected $5$-manifold with $$\mathrm{H}$_2(M_2,$\mathbb{Z}$)=$\mathbb{Z}$/2$\mathbb{Z}$\oplus $\mathbb{Z}$/2$\mathbb{Z}$$,$nM_2$isthe$n$-foldconnectedsumof$M_2$,and$2(S^2\times S^3)$isthetwo-foldconnectedsumof$S^2\times S^3$.